# Product decompositions of quasirandom groups and a Jordan type theorem

Journal of the European Mathematical Society (2011)

- Volume: 013, Issue: 4, page 1063-1077
- ISSN: 1435-9855

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topNikolov, Nikolay, and Pyber, László. "Product decompositions of quasirandom groups and a Jordan type theorem." Journal of the European Mathematical Society 013.4 (2011): 1063-1077. <http://eudml.org/doc/277468>.

@article{Nikolov2011,

abstract = {We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$, then for every subset $B$ of $G$ with $|B|>|G|/k^\{1/3\}$ we have $B^3=G$.
We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs.
On the other hand, we prove a version of Jordan’s theorem which implies that if $k\ge 2$, then $G$ has a proper subgroup of index at most $c_0k^2$ for some constant $c_0$, hence a product-free subset of size at least $|G|/(ck)$. This answers a question of Gowers.},

author = {Nikolov, Nikolay, Pyber, László},

journal = {Journal of the European Mathematical Society},

keywords = {quasirandom groups; product-free sets; word values; finite simple groups; product-free subsets; product decompositions; word maps; character degrees; minimal degree representations},

language = {eng},

number = {4},

pages = {1063-1077},

publisher = {European Mathematical Society Publishing House},

title = {Product decompositions of quasirandom groups and a Jordan type theorem},

url = {http://eudml.org/doc/277468},

volume = {013},

year = {2011},

}

TY - JOUR

AU - Nikolov, Nikolay

AU - Pyber, László

TI - Product decompositions of quasirandom groups and a Jordan type theorem

JO - Journal of the European Mathematical Society

PY - 2011

PB - European Mathematical Society Publishing House

VL - 013

IS - 4

SP - 1063

EP - 1077

AB - We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$, then for every subset $B$ of $G$ with $|B|>|G|/k^{1/3}$ we have $B^3=G$.
We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs.
On the other hand, we prove a version of Jordan’s theorem which implies that if $k\ge 2$, then $G$ has a proper subgroup of index at most $c_0k^2$ for some constant $c_0$, hence a product-free subset of size at least $|G|/(ck)$. This answers a question of Gowers.

LA - eng

KW - quasirandom groups; product-free sets; word values; finite simple groups; product-free subsets; product decompositions; word maps; character degrees; minimal degree representations

UR - http://eudml.org/doc/277468

ER -

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